Computational Stochastic Mechanics of a Simple Bound State

TL;DR

This work formulates and tests a numerical stochastic mechanics framework to compute probability density distributions for quantum-like particles in arbitrary potentials by directly deriving a velocity field from energy and potential via the bridge equation and performing Langevin-like histories. Using the harmonic oscillator as a tractable test case, it assesses discretization effects, convergence, and the influence of energy defect on stability, showing that ground-state-like distributions are recoverable and that delta_E governs the basin of stability and state lifetimes. The results demonstrate the feasibility of predicting structure and reactivity from a classical potential without bootstrapping from a wave function, while highlighting the need for dynamic time-stepping, excited-state velocity fields, and scalable algorithms for broader applications in chemistry, materials, and information sciences.

Abstract

Stochastic mechanics is based on the hypothesis that all matter is subject to universal modified Brownian motion. In this report, we calculated probability density distributions using concepts of stochastic mechanics independent of bootstrapping with a known wave function. We calculate a velocity field directly from the potential and total energy and then use the resultant velocity field to do a stochastic Langevin integration over histories to create the probability density distribution for particle position. Using the harmonic oscillator as a minimally sufficient system for our exposition, we explored the effects of spatial and time discretization on solution noise. We explore the effect of energy defect off of the ground state energy on the velocity field, which dictates how a particle interacts with the background of stochastic fluctuations in position, and describes how repulsive drift (negative defect) and constructive oscillation (positive defect) end a stable state as its basin of stability in the velocity field shrinks with increasing energy defect. The results suggest a pathway for future development of stochastic mechanics as a numerical strategy to describe the structure of physical quantum systems for applications in chemistry, materials and information sciences.

Figures (5)

• Figure 1: (a) Normalized residency histograms (probability densities) after $10^8$ iterations of $\delta t = 0.005$ with $n = 32$, $64$, $128$, and $256$ collocation points. (b) Solution noise measured using equation \ref{['eq:noise']} of the states shown in (a). The points are the mean and shaded area represents the standard deviation of 12 simulations.
• Figure 2: (a) Cumulative solution histograms for $n = 128$ and $\delta t = 0.001$ at iteration $i$ and normalized after $10^8$ iterations. (b) Solution noise was calculated as a function of iteration count for different time step sizes demonstrating that convergence to low-noise solutions may be accelerated using dynamically-sized $\delta t$ which starts large at high noise and decreases with iteration count. The solid line is the mean and shaded region represents the standard deviation across 12 repeated simulations.
• Figure 3: The relationship between converged solution noise at 108 iterations at $n = 128$ as a function of time increment $\delta t$. Points are the averages across 12 simulations whereas the shaded area represents the standard deviation. The dashed line represents a power-law fit to the averaged data of $0.00228 \delta t^2$.
• Figure 4: The effect of energy defect ($\delta E$) on the velocity field changes the characteristic of the particle motion.
• Figure 5: (a) Stability lifetime ($\tau$) as a function of simulation time increment ($\delta t$). (b) $\tau$ as a function of $\delta E$ a precipitous spike in stability at $\delta E = 0$.